8p^2-19=p

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Solution for 8p^2-19=p equation:


Simplifying
8p2 + -19 = p

Reorder the terms:
-19 + 8p2 = p

Solving
-19 + 8p2 = p

Solving for variable 'p'.

Reorder the terms:
-19 + -1p + 8p2 = p + -1p

Combine like terms: p + -1p = 0
-19 + -1p + 8p2 = 0

Begin completing the square.  Divide all terms by
8 the coefficient of the squared term: 

Divide each side by '8'.
-2.375 + -0.125p + p2 = 0

Move the constant term to the right:

Add '2.375' to each side of the equation.
-2.375 + -0.125p + 2.375 + p2 = 0 + 2.375

Reorder the terms:
-2.375 + 2.375 + -0.125p + p2 = 0 + 2.375

Combine like terms: -2.375 + 2.375 = 0.000
0.000 + -0.125p + p2 = 0 + 2.375
-0.125p + p2 = 0 + 2.375

Combine like terms: 0 + 2.375 = 2.375
-0.125p + p2 = 2.375

The p term is -0.125p.  Take half its coefficient (-0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
-0.125p + 0.00390625 + p2 = 2.375 + 0.00390625

Reorder the terms:
0.00390625 + -0.125p + p2 = 2.375 + 0.00390625

Combine like terms: 2.375 + 0.00390625 = 2.37890625
0.00390625 + -0.125p + p2 = 2.37890625

Factor a perfect square on the left side:
(p + -0.0625)(p + -0.0625) = 2.37890625

Calculate the square root of the right side: 1.542370335

Break this problem into two subproblems by setting 
(p + -0.0625) equal to 1.542370335 and -1.542370335.

Subproblem 1

p + -0.0625 = 1.542370335 Simplifying p + -0.0625 = 1.542370335 Reorder the terms: -0.0625 + p = 1.542370335 Solving -0.0625 + p = 1.542370335 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + p = 1.542370335 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + p = 1.542370335 + 0.0625 p = 1.542370335 + 0.0625 Combine like terms: 1.542370335 + 0.0625 = 1.604870335 p = 1.604870335 Simplifying p = 1.604870335

Subproblem 2

p + -0.0625 = -1.542370335 Simplifying p + -0.0625 = -1.542370335 Reorder the terms: -0.0625 + p = -1.542370335 Solving -0.0625 + p = -1.542370335 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + p = -1.542370335 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + p = -1.542370335 + 0.0625 p = -1.542370335 + 0.0625 Combine like terms: -1.542370335 + 0.0625 = -1.479870335 p = -1.479870335 Simplifying p = -1.479870335

Solution

The solution to the problem is based on the solutions from the subproblems. p = {1.604870335, -1.479870335}

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